Question

If \[m[-3\text{ }4]+n[4\text{ }-3]=[10\text{ }-11]\] , then \[3m+7n\] is equal to
A. \[3\]
B. \[5\]
C. \[10\]
D. \[1\]

Answer 1

Hint: To solve this problem, firstly we have to multiply the given matrices with the given constants and then we have to perform the addition operation and after that we will get two equations, we have to solve these two equations by performing some arithmetic operations and then substitute values in the given equation and we will get our required answer.

Complete step by step answer:
A matrix can be defined as a rectangular array of numbers that are generally arranged in rows and columns (it can also be explained as an arrangement of certain quantities). If a matrix is defined as \[m\times n\] means that matrix has \[m\] rows (i.e. horizontal lines) and \[n\] columns (i.e. vertical lines).
The elements of a matrix may be real or complex numbers. If all the elements of a matrix are real, then the matrix is called a real matrix.
Types of matrices are as follows: Row Matrix, Column Matrix, Null Matrix, Square Matrix, Diagonal Matrix, Symmetric Matrix, Skew-Symmetric Matrix, Anti Symmetric Matrix etc.
Let’s discuss addition in matrices:
If we want to add two matrices then both the matrices must have an equal number of columns and rows. The sum of two matrices which has same number of rows and columns can be given as:
\[A+B=\left( \begin{matrix}
   {{a}_{11}} & {{a}_{12}} \\
   {{a}_{21}} & {{a}_{22}} \\
\end{matrix} \right)+\left( \begin{matrix}
   {{b}_{11}} & {{b}_{12}} \\
   {{b}_{21}} & {{b}_{22}} \\
\end{matrix} \right)\]
\[A+B=\left( \begin{matrix}
   {{a}_{11}}+{{b}_{11}} & {{a}_{12}}+{{b}_{12}} \\
   {{a}_{21}}+{{b}_{21}} & {{a}_{22}}+{{b}_{22}} \\
\end{matrix} \right)\]
An addition follows all these given properties in the matrix: The Commutative Law, The associative law, the existence of additive identity and the existence of additive inverse.
We have given in the question as follows:
\[m[-3\text{ }4]+n[4\text{ }-3]=[10\text{ }-11]\]
Now simplifying the matrices, by multiplying it with constant
\[\Rightarrow [-3m\text{ }4m]+[4n\text{ }-3n]=[10\text{ }-11]\]
Adding both the matrices
\[\Rightarrow [-3m+4n\text{ }4m-3n]=[10\text{ }-11]\]
Now, when we will compare two matrices, we will get:
\[\Rightarrow -3m+4n=10\] \[........................(i)\]
\[\Rightarrow 4m-3n=-11\] \[........................(ii)\]
Now we will solve these both equations by multiplying equation \[(i)\] with \[4\] and equation \[(ii)\] with \[3\] :
\[\Rightarrow -12m+16n=40\]
\[\Rightarrow 12m-9n=-33\]
Now we will solve both of these equations by adding both of these equations, we will get:
\[\Rightarrow (-12m+16n+12m-9n)=40-33\]
\[\Rightarrow 7n=7\]
\[\Rightarrow n=1\]
Substitute \[n=1\] in the equation \[(i)\]
\[\Rightarrow -3m+4(1)=10\]
\[\Rightarrow -3m+4=10\]
\[\Rightarrow -3m=10-4\]
\[\Rightarrow -3m=6\]
\[\Rightarrow m=-2\]
Now, we will substitute the values \[n=1\] and \[m=-2\] in \[3m+7n\]
\[\Rightarrow 3m+7n=3(-2)+7(1)\]
\[\Rightarrow -6+7\]
\[\Rightarrow 1\]

So, the correct answer is “Option D”.

Note: A matrix is said to be a symmetric matrix if it follows these given conditions: it has only real Eigenvalues, it is always diagonalizable and it has orthogonal Eigenvectors. And the sum and difference of any two symmetric matrices will always be symmetric but it is not always true in multiplication of matrices.